Answer on Abstract Algebra Question for jeremy

2013-02-06T12:44:25-0500

Question #23892

Let k be any field of characteristic 3, G = S3 and let V be the kG-module ke1 &oplus; ke2 &oplus; ke3/k(e1 + e2 + e3),on which G acts by permuting the ei&rsquo;s. Show that this realizes G as a linear group in GL(V ).

Expert's answer

2013-02-14T08:37:10-0500

Since (123) does not act triviallyon V , the representation homomorphism ϕ : G &rarr; GL(V ) must be injective.Therefore, ϕrealizes G as alinear group in GL(V).